The state Senate is willing to dedicate more to roads and bridges than the $500 million in new taxes approved by the House late Monday night, but still will not come close to the level of investment Governor Deval Patrick has requested, Senate President Therese ­Murray signaled on Tuesday.

The transportation funding bill headed for the Senate floor endorses the same tax increases approved by the House Monday, but would boost transportation spending by redirecting money from other parts of the state budget and imposing a new fee on utility companies. By fiscal 2018, the Senate draft would spend about $126 million more on transportation than the House version, Senate officials said.

This step toward compromise could place Patrick, who has requested an additional $1.9 billion in tax revenue to pay for transportation and education, in the position of either softening his insistence on a tax hike of more than $1 billion (his veto threat was specifically aimed at the House bill, which would be rendered obsolete by the Senate bill), or risk losing what could be his last chance to ­increase spending on what he calls critical investments.

How the Legislature and the governor resolve the tense debate will determine whether MBTA fares and highway tolls increase dramatically, which roads and bridges can get fixed, as well as how much more money Massachusetts taxpayers will have to shell out in taxes each year.

The Senate draft would levy $40 million in new charges on utility companies by requiring them to pay for infrastructure, such as light poles, on state highway right-of-ways. The bill would also divert roughly $80 million in excess funds from an underground storage tank cleanup program, which now flow into the state’s general fund, into the transportation accounts. Like the House, the measure would boost taxes on gas, tobacco, and businesses.

The Senate could vote on the bill as early as Thursday, but liberal senators who agree with Patrick are likely to force a ­delay until early next week.

The Legislature and governor have been engaged in something of a cold war since House and Senate leaders ­announced that their tax plan would come nowhere close to the $1.9 billion Patrick requested earlier this year. By offering Patrick a higher transportation spending figure than the House version, which he has promised to veto, the Senate could allow both Patrick and the House to save face.

“If that bill doesn’t go forward, then the T rates will go up, tolls will go up, and I don’t think anybody wants that to happen,” Murray said Tuesday before the Senate draft was ­released.

The two sides still seem far apart. Asked Tuesday what he would consider acceptable, ­Patrick replied that “there’s a number between where they are and where I am — and I mean really, exactly, between where they are and where I started — where we can make some meaningful improvements in our transportation system and take some first meaningful steps on the education side.”

House Ways and Means chairman Brian Dempsey, a Haverhill Democrat, told report­ers last week that it was “highly unlikely” the House would revisit transportation financ­ing this year if its $500 million preference did not become law. But House officials said Tuesday that a Senate ­increase of a modest size would probably win favor in the lower chamber.

First, though, Murray must muscle the legislation through the Senate, whose membership has changed significantly in the last several years, growing ­noticeably more liberal. Several of those senators side with ­Patrick on tax and transportation policy and are strongly critical of the House transportation bill, which eschews Patrick’s ­income tax hike and sales tax reduction for a range of taxes on gas, tobacco and businesses.

Senator James Eldridge, an Acton Democrat, called the House bill regressive. “I think we need a much stronger revenue package,” he said.

A fellow liberal, Senator ­Sonia Chang-Diaz, Democrat of Boston, blocked a procedural vote Tuesday that would have teed up a Thursday vote.

The Senate planned to take up the House bill Thursday, but the administration has been lobbying for a delay as it seeks to muster the votes needed to sustain the veto.

Murray said Tuesday that the Senate would work through any efforts mounted by Patrick supporters to put off the vote.

“I understand the governor’s asked them to delay the bill as long as they can, but we’ll keep at it until we get to actually taking it up and hopefully pass it,” she said.

Despite leadership efforts, the House vote was eight votes shy of the number needed to override the governor’s threatened veto. The opposition to the bill came from Republicans, whose opposition is rooted in its tax component, as well as liberal Democrats who believe the state should spend more on transportation.

Murray declined to say whether the bill would get the votes needed to override a veto.

While a higher price tag would probably fetch votes from liberal lawmakers, Murray questioned the feasibility of a package in the neighborhood of $1.2 billion, which Patrick had suggested as a halfway figure. “Where does it come from?” she asked of the funding needed to reach that level.

Murray said that the bill Senate leaders will send to the floor will have enough support to pass.

Deep divisions on transportation financing are nothing new on Beacon Hill, which has for decades grappled with how to pay for the Big Dig and the hole it blew in the state’s wallet. Nor is it unusual for feelings among high-level figures to grow personal and hostile.

Lawmakers mocked Governor Mitt Romney in 2006 for proposing to tear down the tollbooths on the Massachusetts Turnpike, deriding it as a political ploy intended to aid Lieutenant Governor Kerry Healey’s gubernatorial campaign. In 2009, Patrick’s transportation chief, James Aloisi, incurred Murray’s ire by mocking her “reform before revenue” mantra as a “meaningless slogan.”

This time through the ­debate, though, dynamics have shifted. Patrick has ruled out a run for a third term, meaning he will not ask the state’s voters again for control of the Beacon Hill purse.

Lawmakers, however, will be on the ballot again in 2014 and have watched colleagues lose their seats after voting for higher taxes.

Several lawmakers said ­Patrick’s request that they show “political courage” rings hollow in light of his own campaign trail language in 2010. That year, Patrick said he did not ­intend to pursue a broad-based tax increase.

If the Legislature sends ­Patrick a bill with a bottom line similar to the Senate’s, said one longtime budget observer, ­Patrick will have achieved “a major victory,” persuading lawmakers to go along with a massive tax increase and taking a stride toward closing the transportation funding gap.

“This would be, in my view, a great success for the governor,” said Michael Widmer, president of the Massachusetts Taxpayers Foundation. “Now, he has put on the table a larger package, but this is still an ­important piece. It’s not as if he doesn’t care about it. So I would hope he would sign it.”